<p>The area bounded by the curve <em>y</em> = <em>f</em>(<em>x</em>), <em>x</em>-axis and the ordinates <em>x</em> = 1 and <em>x</em> = <em>b</em> is (<em>b</em> − 1) sin(3<em>b</em> + 4). Find <em>f</em>(<em>x</em>).</p>
Step-by-Step Solution
Key Concept: Differentiate the area function with respect to b using Leibniz rule: if Area(b) = ∫₁ᵇ f(x)dx, then f(b) = d(Area)/db. Apply product rule to (b−1)sin(3b+4).
<p><strong>Step 1:</strong> Use the Fundamental Theorem of Calculus. If Area = ∫₁ᵇ f(x)dx = (b−1)sin(3b+4), then f(b) = d(Area)/db</p><p><strong>Step 2:</strong> Differentiate (b−1)sin(3b+4) using the product rule: d/db[(b−1)sin(3b+4)]</p><p><strong>Step 3:</strong> Apply product rule: (1)·sin(3b+4) + (b−1)·d/db[sin(3b+4)]</p><p><strong>Step 4:</strong> Evaluate: sin(3b+4) + (b−1)·cos(3b+4)·3</p><p><strong>Step 5:</strong> Simplify: sin(3b+4) + 3(b−1)cos(3b+4)</p><p><strong>Step 6:</strong> Replace b with x (since this must hold for any upper limit): f(x) = sin(3x+4) + 3(x−1)cos(3x+4)</p><p>∴ Answer: <strong>f(x) = sin(3x+4) + 3(x−1)cos(3x+4)</strong></p>
Correct Answer: f(x) = sin(3x+4) + 3(x-1)cos(3x+4)